In this post, we discuss a theorem on tangent of circles. We show that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In the figure below, line m is perpendicular to at . We show that it is a tangent to circle O.

**Theorem**

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

**Given**

Line is perpendicular to at .

**Prove**

Line is tangent to circle

**Proof**

Let be another point on line other than .

Since is perpendicular , triangle is a right triangle with hypotenuse . This means that and must be on the exterior of the circle. Therefore cannot lie on the circle and is the only point of m that is on the circle. So, is a tangent to the circle.

The converse of this statement is also true (see proof).

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